AM 106/206: Applied Algebra Prof. Salil Vadhan Lecture Notes 12 October 20, 2010 • Reading: Gallian Chapters 9 & 10 1 Normal Subgroups • Motivation: { Recall that the cosets of nZ in Z (a+nZ) are the same as the congruence classes modulo n ([a]n) ∗ { These form a group under addition, isomorphic to Zn:[a + b]n = [a + b mod n]n depends only on [a]n and [b]n (and not on the particular choice of coset representatives a and b), so we can define [a]n + [b]n = [a + b]n. { Q: under what conditions on H ≤ G do the left-cosets of H form a group under the operation of G? • Def: A subgroup H of G is normal iff for every a 2 G, aH = Ha. If this holds, we write H C G. • Proposition: For H ≤ G, the following are equivalent: { H C G { for every a 2 G, aHa−1 = H { for every a 2 G, h 2 H, aha−1 2 H. That is, if h 2 H, then all conjugates of h are also in H. • Examples: { Which subgroups of an abelian group are normal? { Which subgroups of S4 are normal? { An C Sn 1 2 Factor Groups • Thm 9.2: If HCG, then the operation (aH)(bH) = abH on the left-cosets of H is well-defined (does not depend on the choice of coset representatives a; b) and forms a group, denoted G=H (called a factor group, the quotient of G by H, or \G mod H"). • Proof: • Examples: { Z=nZ { Sn=An { (Z3 × Z5)=(Z3 × f0g) { S4=H where H is the normal subgroup of size 4. { Zn=hai { R=Z 3 Homomorphisms • Def: For groups G, H, and mapping ' : G ! H is a homomorphism if for all a; b 2 G, we have '(ab) = '(a)'(b). { Note: we don't require that ' is one-to-one or onto! • Def: For a homomorphism ' : G ! H, { the image of ' is Im(') = '(G) = f'(g): g 2 Gg ≤ H. { the kernel of ' is Ker(') = fg 2 G : '(g) = "g C G. • Thm 10.3: If ' : G ! H is a homomorphism, then G=Ker(') =∼ Im('). Picture: 2 • Examples: Domain Range Mapping Homo.? Image Kernel Z Zn x 7! x mod n Zn Zd x 7! x mod d n n R R x 7! Mx, M a matrix Z × Z Z (x; y) 7! xy Sn {±1g σ 7! sign(σ) ∗ 2πix R C x 7! e Z3 × Z5 Z3 (x; y) 7! x G G=N, where N C G g 7! gN • Properties of Homomorphisms: 1. '("G) = "H . 2. '(a−1) = '(a)−1. 3. order('(a)) divides order(a). • Properties of Images: 1. '(G) is a subgroup of H. 2. G cyclic ) '(G) cyclic. 3. G abelian ) '(G) abelian. • Properties of Kernels: 1. Ker(') is normal subgroup of G. { Can prove that K is normal by finding a homomorphism ' s.t. Ker(') = K. 2. '(a) = '(b) , b−1a 2 Ker(') , aKer(') = bKer('). 3. ' injective (one-to-one) if and only if Ker(') = f"g. • Proof of Thm 10.3 (G=Ker(') =∼ Im(')): 3.